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30.2 Seismic Random-Excitation Fields

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30.2.1 Power Spectral Density of Spatially Varying Ground Acceleration

For multiple excitation problems, the PSD matrices of ground acceleration excitations have the form

Sxx ðivÞ ¼

SX€ 1

X€ 1 ðivÞ SX€ 1

X€ 2 ðivÞ · · · SX€ 1

X€ N ðivÞ

SX€ 2

X€ 1 ðivÞ SX€ 2

X€ 2 ðivÞ · · · SX€ 2

X€ N ðivÞ

· · · · ·· ·· · · · ·

SX€ N

X€ 1 ðivÞ SX€N

X€ 2 ðivÞ · · · SX€N

X€ N ðivÞ

6666664

7777775

ð30:50Þ

in which

SX€ k

X€ l ðivÞ ¼ rklðivÞ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

SX€ k ðvÞSX€ l ðvÞ

ð30:51Þ

rklðivÞ ¼ lrklðivÞlexp½iuklðvÞ􀀉 ð30:52Þ

and rklðivÞ is the acceleration coherence function between the kth and lth supports. Its norm must satisfy

the relation lrklðivÞl # 1: The values of SX€ k ðvÞ and SX€ l ðvÞ in Equation 30.51 can be different due to local

effects. However, earthquake records show that the structural responses due to the difference between

SX€ k ðvÞ and SX€ l ðvÞ are in general of relatively low significance. The factor exp½iuklðvÞ􀀉 expresses the wave

passage effect, which can be further expressed as

exp½iuklðvÞ􀀉 ¼ exp½2ivdL

kl=vapp􀀉 ð30:53Þ

in which dkl is the horizontal distance between the two supports; dL

kl is the projection of dkl along the

propagation direction of the seismic waves; and vapp is the apparent velocity of seismic waves along the

surface. Assume that the time lags between the supports and the origin of the reference coordinate system

are t1; t2; …; tN ; respectively. Without losing generality, let tl $ tk (when l . k). Then dL

kl=vapp ¼ tl 2 tk;

and so Equation 30.53 becomes

exp½iuklðvÞ􀀉 ¼ exp½ivðtk 2 tlÞ􀀉 ð30:54Þ

The factor lrklðivÞl reflects the incoherence effect (Kiureghian and Neuenhofer, 1992). Some

mathematical models of rklðivÞ have been established based on practical earthquake records, which will

be shown in the next section. Using Equation 30.51 to Equation 30.54, Equation 30.50 can be written as

SðivÞ ¼ BpJRJB ð30:55Þ

in which

B ¼ diag½expð2ivt1Þ; expð2ivt2Þ; …; expð2ivtN Þ􀀉 ð30:56Þ

J ¼ diag

ffiffiffiffi

SX€ 1

;

ffiffiffiffi

SX€ 2

; …;

ffiffiffiffiffi

SX€ N

h q i

ð30:57Þ

Seismic Random Vibration of Long-Span Structures 30-11

R ¼

1 lr12l · · · lr1N l

lr21l 1 · · · lr2N l

· ·· ·· · · · · · · ·

lrN1l lrN2l · · · 1

6666664

7777775

ð30:58Þ

30.2.2 Several Coherence Models

A number of coherence models have been established based on practical earthquake records. Some of

them are outlined below.

30.2.2.1 Feng – Hu Model

lrklðv; dklÞl ¼ exp½2ðr1v þ r2Þdkl􀀉 ð30:59Þ

in which r1 and r2 are the coherence parameters. According to the Hai-Cheng earthquake

records (China) and the Niigata earthquake records (Japan), the values of these parameters are

(Feng and Hu, 1981):

Hai-Cheng: r1 ¼ 2 £ 1025 sec/m, r2 ¼ 88 £ 1024 m21

Niigata: r1 ¼ 4 £ 1024 sec/m, r2 ¼ 19 £ 1024 m21

30.2.2.2 Harichandran – Vanmarcke Model

lrklðv; dklÞl ¼ A exp 2

auðvÞ ð1 2 A þ aAÞ

􀀒 􀀓

þ ð1 2 AÞexp 2

auðvÞ ð1 2 A þ aAÞ

􀀒 􀀓

ð30:60Þ

in which

uðvÞ ¼ K

􀀑

1 þ ðv=v0Þb􀀜21=2 ð30:61Þ

According to the acceleration records of the SMART-1 array (Harichandran and Vanmarcke, 1986), the

parameters in Equations 30.60 and Equations 30.61 are: A ¼ 0:736; a ¼ 0:147; K ¼ 5210;

v0 ¼ 6:85 rad/sec, b ¼ 2:78:

30.2.2.3 Loh – Yeh Model

lrklðv; dklÞl ¼ exp 2a

vdkl

2pvapp

" #

ð30:62Þ

in which a is the wave-number of the seismic waves. According to the 40 acceleration records of the

SMART-1 array (Loh and Yeh, 1988), a ¼ 0:125 is proposed. Parameters vapp and dkl have been explained

above.

30.2.2.4 Oliveira – Hao – Penzien Model

lrklðv; dklÞl ¼ expð2b1dL

kl 2 b2dT

klÞexpð2

􀀑

ffiffiffiffi

2 a2

ffiffiffiffi

q 􀀜

ðv=2pÞ2Þ ð30:63Þ

in which ai ¼ ð2pai=vÞ þ ðvbi=2pÞ þ ci: dL

kl and dT

kl are the projections of dkl along the propagation

direction of the seismic waves and along its normal direction, respectively. Based on 17 acceleration

records of the SMART-1 array (Oliveira et al., 1991), parameters b1; b2; a1; a2; b1; b2; c1, and c2 are given

for each of these records.

30-12 Vibration and Shock Handbook

30.2.2.5 Luco–Wong Model

lrkll ¼ exp 2

avdkl

􀀏 􀀐2

" #

ð30:64Þ

in which a is the coherence factor, dkl is the horizontal distance between the kth and lth supports and vs is

the shear wave velocity (Luco and Wong, 1986).

30.2.2.6 Qu–Wang–Wang Model

T. Qu, J. Wang, and Q. Wang (QWW) (Qu et al., 1996) proposed the following model

lrkll ¼ exp 2aðvÞdbðvÞ kl

h i

ð30:65Þ

in which

aðvÞ ¼ a1v2 þ a2; bðvÞ ¼ b1v þ b2 ð30:66Þ

a1 ¼ 0:00001678; a2 ¼ 0:001219; b1 ¼ 20:0055; b2 ¼ 0:7674: This model was established based on the

statistics of dozens of records from four closely located arrays with SMART-1 as the leading one. This

model has given reasonable results in some applications in China.

30.2.3 Generation of Ground Acceleration Power Spectral Density Curves

from Acceleration Response Spectrum Curves

In order to carry out the seismic random vibration analysis of important long-span bridges, the local

seismic motion PSD (usually the acceleration PSD) must be established by specialists. For bridges of less

importance, however, such seismic acceleration PSD can be derived from the local ARS curve. Two

methods to perform the transformation are given below.

30.2.3.1 Kaul Method

Kaul (1978) proposed an approximate transformation method that consists of the two equations

Sðv0Þ ¼

4jR2

a ðj; v0Þ

pv0r2 ð30:67Þ

r2 ¼ 2 ln 2

v0Ts

ln p

􀀏 􀀐21 􀀒 􀀓

ð30:68Þ

in which Raðj; v0Þ is the ARS curve of the seismic absolute acceleration when the damping ratio is j

and the natural angular frequency is v0 (usually j takes 0.05); Sðv0Þ is the equivalent PSD curve

corresponding to the given ARS curve; Ts is the seismic duration; and p is the probability of the peaks that

do not cross with the given positive or negative barriers — usually p ¼ 0:85 is assumed.

30.2.3.2 Iteration Scheme

An iteration-based scheme for the transformation has been proposed (Sun and Jiang, 1990), which

produces more accurate equivalent PSD curve than the Kaul method. For a given ARS curve Raðj; v0Þ; in

order to transform it into an equivalent PSD curve SðvÞ; specify its initial values S0ðviÞ ¼ Si; for

i ¼ 1; 2; …; N: Then the standard deviation of the acceleration response for a SDOF system can be

computed from

s0ðj; v0Þ ¼

ð1

SðvÞ

1 þ 4j2ðv=v0Þ2

ð1 2 ðv=v0Þ2Þ2 þ 4j2ðv=v0Þ2 dv

" #1=2

ð30:69Þ

The peak factor r is

r ¼

ffiffiffiffiffiffiffiffiffiffiffi

2 lnðvTsÞ

þ 0:5772

. ffiffiffiffiffiffiffiffiffiffiffi

2 lnðvTsÞ

ð30:70Þ

Seismic Random Vibration of Long-Span Structures 30-13

in which Ts is the period of the strong earthquake portion. The average zero-crossing rate is

approximately v ¼

ffiffiffiffiffiffiffi

l2=l0 p 􀀋

p < v0=p:

The acceleration responses computed by means of Equation 30.69 and Equation 30.70 are

Amðj; v0Þ ¼ rs0ðj; v0Þ ð30:71Þ

The percentage errors between Raðj; v0Þ and Amðj; v0Þ can be computed from

Eðv0Þ ¼

lRaðj; v0Þ 2 Amðj; v0Þl

Raðj; v0Þ £ 100% ð30:72Þ

Compute Eðv0Þ for each frequency. If Eðv0Þ is found to be greater than the given tolerance 1 for at least

one frequency, modify all PSD values according to the following equation

Skþ1ðviÞ ¼ Sk ðviÞR2

a ðj; viÞ

􀀋

mðj; viÞ; i ¼ 1; 2; …; N ð30:73Þ

and then repeat the computations of Equation 30.69 to Equation 30.72. The above process is continued

until Equation 30.72 is satisfied at all frequencies.

30.2.4 Seismic Equations of Motion of Long-Span Structures

For long-span structures subjected to differential ground motion, the equations of motion in the global

coordinate system (assumed to be fixed to the center of the Earth) can be written in partitioned form as

Ms Msb

sb Mb

" #

x€s

x€b

( )

Cs Csb

sb Cb

" #

x_s

x_b

( )

Ks Ksb

sb Kb

" #

( )

ð30:74Þ

in which the subscript m represents the master DoF, that is, the support displacements, while the

subscript s represents the slave DoF. The absolute displacement vector xs can be decomposed into the

two parts

xs ¼ ys þ yr ð30:75Þ

where ys is the quasi-static displacement vector (Clough and Penzien, 1993), which satisfies

ys ¼ 2K21

s Ksbxb ð30:76Þ

Substituting Equation 30.75 and Equation 30.76 into Equation 30.74 gives

Msy€r þ Csy_r þ Ksyr ¼ MsK21

s Ksbx€b þ ðCsK21

s Ksb 2 CsbÞx_b ð30:77Þ

It should be pointed out that Equation 30.77 cannot be reduced to the conventional Equation 30.24

when xb represents uniform ground displacements (Clough and Penzien, 1993). This is because Equation

30.74 assumes the damping forces to be proportional to the absolute velocity vector {x_T

s ; x_Tb

}T: In order to

avoid this inconsistency, the damping forces should be assumed to be proportional to the relative velocity

vector {y_T

r ; 0}T in Equation 30.74. This leads to the equations

Msy€r þ Csy_r þ Ksyr ¼ MsK21

s Ksbx€b ð30:78Þ

for uniform ground motion

x€b ¼ ebx€g ð30:79Þ

Note that the following rigid displacement condition is satisfied:

Ks Ksb

sb Kb

" #

( )

ð30:80Þ

30-14 Vibration and Shock Handbook

Its second half gives

Ksbeb ¼ 2Kses ð30:81Þ

Substituting Equation 30.79 into Equation 30.78 and using Equation 30.81 gives Equation 30.24.

30.2.5 Seismic Waves and Their Geometrical Expressions

Seismic waves can be divided into body waves and

surface waves. Body waves include longitudinal

waves (or pressure waves, primary waves or P

waves) and transverse waves (or shear waves,

secondary waves or S waves). Surface waves include

Rayleigh waves and Love waves. For P waves, the

soil particles move parallel to the traveling direction

of waves; for S waves, however, their motion

is normal to the wave traveling direction

(see Figure 30.4). For horizontal shear waves

(SH waves), all particles move horizontally.

For vertical shear waves (SV waves), all particles

move vertically.

Assume that both x and y axes lie in the

horizontal plane. The angle between axis x and the

horizontal traveling direction of these waves is b;

as shown in Figure 30.5. Thus, the acceleration

components along the coordinate axes can be

expressed by the components parallel, or normal,

to the wave traveling direction, that is, for P waves

x€i ¼ u€ i cos b; y€i ¼ u€ i sin b;

z€i ¼ 0 ð30:82Þ

for SH waves

x€i ¼ 2u€ j sin b; y€i ¼ u€ j cos b;

z€i ¼ 0 ð30:83Þ

and for SV waves

x€i ¼ 0; y€i ¼ 0; z€i ¼ u€ k ð30:84Þ

In the equations of motion of an multi-DoF system under uniform ground excitations, that is, Equation

30.24, e is the index vector of inertia forces. Its mathematical expressions for the different waves are:

for P waves

e ¼ ex cos b þ ey sin b ð30:85Þ

for SH waves

e ¼ 2ex sin b þ ey cos b ð30:86Þ

for SV waves

e ¼ ez ð30:87Þ

Clearly, for P waves, e ¼ ex when b ¼ 0 and e ¼ ey when b ¼ 908:

β x

FIGURE 30.4 Particle motion directions for P and S

waves.

üj

üi

FIGURE 30.5 Transform of ground acceleration

components.

Seismic Random Vibration of Long-Span Structures 30-15

If a structure has N supports, its ground acceleration excitations along the wave traveling direction can

be expressed by the N-dimensional vector

u€ b ¼ {u€ 1; u€ 2;…; u€N}T ð30:88Þ

The m-dimensional ground acceleration vector in Equation 30.78 is

x€b ¼ {x€1; x€2;…; x€m}T ð30:89Þ

The relation between these two vectors is

x€b ¼ EmNu€ b ð30:90Þ

in which EmN is a m £ N block-diagonal matrix

EmN ¼ diag½eb ; eb ; …; eb 􀀉m£N ð30:91Þ

If only three translations are considered for each support, then m ¼ 3N and each submatrix eb

would be

cos b

sin b

8>><

>>:

9>>=

>>;

;

2sin b

cos b

8>><

>>:

9>>=

>>;

and

8>><

>>:

9>>=

>>;

for the P, SH and SV waves, respectively.

Using Equation 30.90, Equation 30.76 and Equation 30.78 can be rewritten as

ys ¼ 2K21

s KsbEmN ub ð30:92Þ

Msy€r þ Csy_r þ Ksyr ¼ MsK21

s KsbEmNu€ b ð30:93Þ

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